Where due to a similar mapping with the one from K-S theorems, but proven by Runge and Gross [1]}, the state \(\Psi \)  is an unique functional of density: \(|\Psi \rangle \equiv |\Psi [\rho (\vec{r},t)]\rangle \) . \(\hat{H}\)  is the Hamiltonian operator. A set of hydrodinamical equations can be derived from the variational principle of least action [1]}:
\(\partial _t\rho (\vec{r},t)+\nabla \vec{j}(\vec{r},t)=0\\\partial _t\vec{j}(\vec{r},t)=\vec{P}[\rho ](\vec{r},t)\\\vec{P}[\rho ](\vec{r},t)=-i\langle \psi [\rho ]|[\hat{j},\hat{H}]|\psi [\rho ]\rangle \) 
